We present theoretical and observational studies of non-Gaussian
fluctuations in the cosmic microwave background (CMB) radiation
anisotropy.
We use the angular bispectrum and trispectrum, the
harmonic transform of the angular three- and four-point correlation functions.
If the primordial fluctuations are non-Gaussian, then this non-Gaussianity
will be apparent in the CMB sky.

Theory of the primordial CMB angular bispectrum

Non-linearity in inflation produces the primordial non-Gaussianity.
We predict the primary angular bispectrum from inflation
down to arcminutes scales, and forecast how well we can measure
the primordial non-Gaussian signal.
In addition to that, secondary anisotropy sources in the low-redshift
universe also produce non-Gaussianity, so do foreground emissions from
extragalactic or interstellar microwave sources.
We study how well we can measure these non-Gaussian signals, including the
primordial signal, separately.
We find that when we can compute the predicted form of the bispectrum,
it becomes a ``matched filter'' for finding non-Gaussianity in the data,
being very powerful tool of measuring weak non-Gaussian signals and of
discriminating between different non-Gaussian components.
We find that slow-roll inflation produces too small bispectrum
to be detected by any experiments; thus, any detection strongly constrains
this class of models.
We also find that the secondary bispectrum from coupling between the
Sunyaev--Zel'dovich effect and the weak lensing effect, and
the foreground bispectrum from extragalactic point sources,
give detectable non-Gaussian signals on small angular scales.

Measurement of the angular bispectrum on the COBE/DMR sky maps

We test Gaussianity of the COBE DMR sky maps, by measuring all
the modes of the angular bispectrum down to the DMR beam size.
We compare the data with the simulated Gaussian realizations, finding
no significant signal of the bispectrum on the mode-by-mode basis.
We also find that the previously reported detection of the bispectrum
is consistent with a statistical fluctuation.
By fitting the theoretical prediction to the data for the primary
bispectrum, we put a constraint on non-linearity in inflation.
Simultaneously fitting the foreground bispectra, which are estimated from
interstellar dust and synchrotron template maps, shows that neither
dust nor synchrotron emissions contribute significantly to the bispectrum
at high Galactic latitude.
We thus conclude that the angular bispectrum finds no significant
non-Gaussian signals in the DMR data.

Measurement of the angular trispectrum on the COBE/DMR sky maps

We present the first measurement of the angular
trispectrum on the DMR sky maps, further testing Gaussianity
of the DMR data.
By applying the same method as used for the bispectrum to the DMR data,
we find no significant non-Gaussian signals in the trispectrum.
Therefore, the angular bispectrum and trispectrum show that the
DMR sky map is comfortably consistent with Gaussianity.

What is in the future?

The methods that we have developed in this thesis can readily be
applied to the MAP data, and will enable us to pursue non-Gaussian
CMB fluctuations with the unprecedented sensitivity.
We show that high-sensitivity measurement of the CMB bispectrum and
trispectrum will probe the physics of the early universe as well as
the astrophysics in the low-redshift universe, independently of the CMB
power spectrum.